GOE statistics on the moduli space of surfaces of large genus

TL;DR

This paper investigates the statistical distribution of Laplace eigenvalues on large genus hyperbolic surfaces, demonstrating GOE statistics emerge in a specific double limit using Mirzakhani's integration formula.

Contribution

It introduces a new linear statistic for hyperbolic surfaces and proves GOE statistics appear in the double limit of large genus and high energy, extending understanding of spectral statistics.

Findings

Variance of the linear statistic matches GOE predictions

GOE statistics are recovered in the double limit

Mirzakhani's formula is crucial for the proof

Abstract

For a compact hyperbolic surface, we define a smooth linear statistic, mimicking the number of Laplace eigenvalues in a short energy window. We study the variance of this statistic, when averaged over the moduli space $\mathcal M_g$ of all genus $g$ surfaces with respect to the Weil-Petersson measure. We show that in the double limit, first taking the large genus limit and then the high energy limit, we recover GOE statistics. The proof makes essential use of Mirzakhani's integration formula.